Short answer: The two disciples of John the Baptist seem to be Andrew (per John 1:40) and John the Apostle.
One key point in trying to identify the two disciples of John the Baptist is that, in writing the fourth Gospel, John the Evangelist/Apostle never names himself. As Gill says, he “always chooses to conceal himself”.
The only point in the fourth Gospel at which “Zebedee’s sons” are mentioned is in the last chapter, days after the resurrection. There in John 21:2 they’re noted as two of the only seven disciples Jesus appears to.
John, in his Gospel several times, simply identifies himself as the one “whom Jesus/he loved”:
-At the last supper [i.e. reclining on Jesus was the one “whom Jesus loved”) John 13:23 (NASB)
-At the foot of the cross (i.e. Jesus’ mother is with “the disciple whom he loved”) John 19:26 (NASB)
-Near the empty tomb with Mary Magdalene running to Peter and the “other disciple whom Jesus loved” John 20:2 (NASB)
-As Peter turns and sees “whom Jesus loved following them; the one who also had leaned back on His bosom at the supper” John 21:20 (NASB)
Matthew and Mark’s Gospels: While John in his Gospel chooses to not name himself, the Gospels of Matthew and Mark clearly do name him. In those two, back when the first disciples are being called, Matthew and Mark note two pairs of brothers being called. First are Simon Peter and Andrew, then the brothers James and John.
[Matthew 4:18-22 (NASB), Mark 1:16-20 (NASB)]
John’s Gospel vs. Matthew and Mark’s:
While Matthew and Mark’s Gospels, on the first day of disciple callings, has the two pairs of brothers named, John’s has only two disciples actually named. There, while Andrew is “one of the two who heard John” the Baptist, the other who heard him simply isn’t named. Instead that day it only has Simon Peter hearing of Jesus from his brother Andrew. John 1:35-42 (NASB)
It isn’t until the next day that any other disciple is named, and that’s Philip.
John the Evangelist, in writing of Jesus telling Andrew (and another) to “Come and see”, appears to be concealing his own identity for the first time; he does that later quite a bit. For the first day of callings he only names the brothers Andrew and Simon Peter; he doesn’t name himself or his brother. At this point he only mentions himself as “one of the two”. In subsequent chapters he’ll write of himself as the one “whom Jesus/he loved”.